2. Why do we need statistics in
analytical chemistry?
• Scientists need a standard format to
communicate significance of experimental
numerical data.
• Objective mathematical data analysis
methods needed to get the most information
from finite data sets
• To provide a basis for optimal experimental
design.
3. What Does Statistics Involve?
• Defining properties of probability
distributions for infinite populations
• Application of these properties to
treatment of finite (real-world) data sets
• Probabilistic approaches to:
– Reporting data
– Data treatment
– Finite sampling
– Experimental design
4. Some Useful Statistics Terms
• Mean – Average of a set of values
• Median – Mid-point of a set of values.
• Population – A collection of an infinite munber of
measurements. N infinity
• Sample – A finite set of measurements which
represent the population.
• True value (true mean)- (m), mean value for the
population.
• Observed Mean –(x), mean value of the sample set
5. Accuracy and Precision:
Is There a Difference?
• Accuracy: degree of agreement between
measured value and the true value.
• Absolute true value is seldom known
• Realistic Definition: degree of agreement
between measured value and accepted true
value.
6. Precision
• Precision: degree of agreement between
replicate measurements of same quantity.
• Repeatability of a result
• Standard Deviation
• Coefficient of Variation
• Range of Data
• Confidence Interval about Mean Value
8. Determinate Errors
Are They Systematic?
• Determinate Errors:
• Determinable and either avoided or
corrected.
• Constant errors
• Uncalibrated weights
• Burets- volume readings can be corrected
• Concentration variation with temperature
9. Indeterminate Errors
Are They Random?
• Indeterminate Errors-
– accidental or random errors
• Represent the experimental uncertainty that
occurs in any measurement.
– Small difference on successive measurements
• Random Distribution
• Mathematical Laws of Probability
• Normal distribution or Gaussian Curve
11. A Review of Significant Figures
How many significant figures in the following
examples?
• 0.216 90.7 800.0 0.0670 500
• ((35.63 * 0.5482 * 0.05300)/1.1689)*100%
• 88.5470578%
• 88.55%
• ((97.7/32.42)*100.0)+36.04)/687
• 0.4911
12. Ways of Expressing Accuracy
• Absolute Errors: difference between true
value and measured value
• Mean Errors: difference between true
value and mean value
• Relative Error: Absolute or Mean Errors
expressed as a percentage of the true value
((m-x)/m)*100 = % Relative Error
• Relative Accuracy: measured or mean
value expressed as a percentage of true
value
((x/m)*100 = % Relative Accuracy
13. Standard Deviation
The Most Important Statistic
• Standard Deviation s of an intinite set of
experimental data is theoretically given by
s = S(xi – m)2/N
• xi = individual measurement
m = mean of infinite number of
measurements (true value)
• N = number of measurements
14. Standard Deviation of a Finite Set
of Experimental Data
• Estimated Standard Deviation, s (N < 30)
• s = (S(xi – x)2/(N-1))
• For finite sets the precision is represented
by s.
• Standard deviation of the mean smean
• Smean = s/N
• Relative standard deviation rsd: or
coefficient of variation
• (s/mean)*100 = % rsd
20. Propagation of Errors
Not Just Additive
Computation Determinate Indeterminate
(Random)
Add/Subtract
R = A+B-C
ER = EA+ EB-EC sR
2 = sA
2+ sB
2+sC
2
sR =sA
2+ sB
2+sC
2
Multiply/Divide
R = AB/C
ER= EA+ EB- EC
R A B C
(sR/R)2 =(sA/A)2+
(sB/B)2+(sc/C)2
General
R = f(A,B,C,…)
21. Control Charts
• Quality control chart: time plot of a
measured quantity assumed to be constant.
• Inner and Outer control limits
• Inner control limit: 2s (1/20)
• Outer control limit: 2.5s (1/100) or
3s(1/500)
23. Confidence Limit
How sure are you?
• Confidence Limit = x ± ts/N
t statistical factor that depends on the number
of degrees of freedom
degrees of freedom = N-1
Values of t at different confidence levels and
degrees of freedom are located in table 3.1
25. Tests of Significance
Is there a difference?
• The F Test
• Designed to indicate whether there is a
difference between two methods.
• F = s1
2/s2
2 degrees of freedoms 1 and 2
If calculated F value exceeds a tabulated F
value at a selected confidence level, then
there is a significant difference between the
variances of the two methods.
29. Tests of Significance
Is there a difference?
• Comparison of the Means of Two Samples
• ±t = ((x1-x2)/sp) (N1N2/(N1+N2))
• pooled standard deviation: sp
• sp = (S(xi1-x1
)2+S(xi2-x2)2+…+S(xik-xk)2/(N-k))
30. Rejection of a Result:
The Q Test
• The Q test is used to determine if an
“outlier” is due to a determinate error. If it
is not, then it falls within the expected
random error and should be retained.
• Q = a/w
• a = difference between “outlier” and nearest
sorted result
• w = range of results.
40. Peak Area vs Mole % Isooctane
PA = 2.0925Mole% + 0.2567
R2
= 0.9877
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0 0.5 1 1.5 2
Mole % Isooctane
Peak
Area
41. Detection Limits
There Is No Such Thing as Zero
• All instrumental methods have a degree of noise
associated with the measurement that limits the
amount of analyte that can be detected.
• Detection Limit is the lowest concentration level
that can be determined to be statistically different
from an analyte blank.
• Detection Limit is the concentration that gives a
signal three times the standard deviation of the
background signal.
